166 9 Fundamental Theorem of Asset Pricing
Proof.SinceS is locally bounded, there is a sequenceαn→+∞and an
increasing sequence of stopping timesTn→∞so that on [[0,Tn]] the process
Sis bounded byαn. We replaceSby
S ̃=S (^1) [[ 0,T 1 ]]+
∑
n≥ 12 −n1
αn+αn+1( (^1) ]]Tn,Tn+1]]·S),
S ̃is bounded and satisfies(NFLVR)since the outcomes of admissible
integrands are the same forSandS ̃. A martingale measure forS ̃is a local
martingale measure forSand therefore the corollary follows from the main
theorem. The proof of the necessity of the condition(NFLVR)is proved in
the same way as in the Theorem 9.1.1.
Remark 9.4.3.The necessity of the condition(NFLVR)and Theorem 9.4.2
show that ifSis a locally bounded local martingale then the setC 0 is Fatou
closed.
We now proceed with the proof of Theorem 9.4.2. The bounded semi-
martingaleS will be assumed to satisfy the property(NFLVR).Wetake
a sequencehn∈C 0 ,hn≥−1andhn→ha.s.; we have to showh∈C 0.
This is the same as showing that there is af 0 ∈K 0 withf 0 ≥h.Foreach
nwe takegn∈K 0 such thatgn≥hn. The sequencegnis not necessarily
convergent and even if it were, this does not give good information about the
sequence of integrands used to constructgn. To overcome this difficulty we
introduce a maximal element (compare Remark 9.4.5 below). DefineDas the
setD={f|there is a sequenceKnof 1-admissible integrands such that
(Kn·S)∞→fa.s. andf≥h}.
Lemma 9.4.4.The setDis not empty and contains a maximal elementf 0.
Proof.Dis not empty. IndeedDcontains an elementgthat dominatesh.
To see this we takegnas above and apply Lemma 9.8.1. Next observe that
the setDis bounded inL^0 since it is contained in the closure of the set
{(H·S)∞|H1-admissible}which is bounded by Corollary 9.3.4. The setD
is clearly closed for the convergence in probability. We now apply the well-
known fact that a bounded closed set ofL^0 contains a maximal element. For
completeness we give a proof. We will use transfinite induction. Forα=1
take an arbitrary elementf 1 ofD.Ifαis of the formα=β+1 and iffβ
is not maximal then choosefα≥fβ;P[fα>fβ]>0andfα∈D.Ifαis
a countable limit ordinal thenα= limβnwhereβnis increasing toα.The
sequencefβn, is increasing and converges to a functionfαfinite a.s. (Dis
bounded!). In this way we construct for each countable ordinal the variable
fα.SinceE[exp(−fα)] is well-defined and form a decreasing “long sequence”,
this sequence has to become eventually stationary, say at a countable ordinal
α 0 .Byconstructionf 0 =fα 0 is maximal.